. . Attainment of bounds . .
(a) Bounds for the number of real roots The polynomial P x i x ( ) = )( )( )( )( has the zeros − , , , , i i . The polynomial, in the customary form is x .This polynomial P x ( ) has sign changes, namely at fourth and zeroth powers. Moreover, P - - has one sign change. By our Descartes rule, the number of positive zeros of the polynomial P x ( ) cannot be more than ; the number of negative zeros of the polynomial P x ( ) cannot be more than .
Clearly and are positive zeros, and − is the negative zero for the polynomial, x and hence the bounds for positive zeros and the bound for negative zeros are attained. We note that i and − i are neither positive nor negative. We know ( )( )( )( i x is a polynomial with roots − , i i . The polynomial, say P x ( ), in the customary form is x This polynomial P x ( ) has no sign change and P has sign changes.
By Descartes rule, the polynomial P x ( )cannot have more than positive zeros and the number of negative zeros of the polynomial P x ( ) cannot be more than . As another example, we consider the polynomial. C x C x C x C + − + − This is the expansion of ( − . This polynomial has n changes in coefficients and P has no change of sign in coefficients.
This shows that the number of positive zeros of the polynomial cannot be more than n and the number of negative zeros of the polynomial cannot be more than . The statement on negative zeros gives a very useful information that the polynomial has no negative zeros. But the statement on positive zeros gives no good information about the positive zeros, though there